Initial conditions estimation for improving forecast accuracy in exponential smoothing

In this paper we analyze the importance of initial conditions in exponential smoothing models on forecast errors and prediction intervals. We work with certain exponential smoothing models, namely Holt’s additive linear and Gardner’s damped trend. We study some probability properties of those models, showing the influence of the initial conditions on the forecast, which highlights the importance of obtaining accurate estimates of initial conditions. Using the linear heteroscedastic modeling approach, we show how to obtain the joint estimation of initial conditions and smoothing parameters through maximum likelihood via box-constrained nonlinear optimization. Point-wise forecasts of future values and prediction intervals are computed under normality assumptions on the stochastic component. We also propose an alternative formulation of prediction intervals in order to obtain an empirical coverage closer to their nominal values; that formulation adds an additional term to the standard formulas for the estimation of the error variance. We illustrate the proposed approach by using the yearly data time-series from the M3-Competition.